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Computable Euclid

Proposition 9

Theorem

If a line AB  is bisected at point C, and divided into two unequal parts at point D, then AD 2 + DB 2 = 2 AC 2 + 2 CD 2.

Commentary

1. Let AB  be the given line segment. Let C be the midpoint of AB , and let D be any point on CB .
2. Construct a line CE  that is perpendicular to AB  and has the same length as AC . Connect AE  and EB .
3. Find a point F on EB  such that CDFG is a rectangle with G on CE . Connect AF .
4. Five isosceles right triangles are constructed: ACE, BCE, AEB, EGF and FDB. Two right triangles that are not isosceles are constructed: ADF and AEF.
5. Apply the Pythagorean Theorem to the right triangles so: in ADF, AF 2 = AD 2 + DF 2, in AEF, AF 2 = AE 2 + EF 2, in ACE, AE 2 = AC 2 + CE 2 and in EGF, EF 2 = EG 2 + GF 2. With substitution of lines with the same length, the following equation is obtained: AD 2 + DB 2 = 2 AC 2 + 2 CD 2.
6. This geometric relationship is similar to the next proposition, Book 2 Proposition 10, and can be expressed algebraically as x + y2 + x - y2 = 2 x2 + 2 y2 (where AC  = CE  = x and CD  = y). This algebraic relationship is an interpretation of a commonly used polynomial identity.

Original statement

ἐὰν ϵὐθϵῖα γραμμὴ τμηθῇ ϵἰς ἴσα καὶ ἄνισα, τὰ ἀπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων τϵτράγωνα διπλάσιά ἐστι τοῦ τϵ ἀπὸ τῆς ἡμισϵίας καὶ τοῦ ἀπὸ τῆς μϵταξὺ τῶν τομῶν τϵτραγώνου.

English translation

If a straight line is cut into equal and unequal segments, the squares on the unequal segments of the whole are double of the square on the half and of the square on the straight line between the points of section.


Computable version


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